I don’t think that requiring the desktop app to be installed would be a great benefit.
Some users, mainly in corporate environment, are not allowed to install arbitrary executables on their computer. I’m currently in one of those environments, and I can use BW because it just takes the form of a browser extension which my company doesn’t enforce any control on.
Though, I do feel like allowing the desktop app and the extension to talk if they’re both installed would be very nice. That would allow to unlock once and use both apps for example.
There are already some feature requests related to this, see: